12 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA Two families of pseudo-distances ( da)a 6 T , (d^)^ e L on the same set E are said to be topologically equivalent if they define the same topology on E. (12.4.5) If (Wa)a6i is any family of pseudo-distances on a set E, there exists a topologically equivalent family (d^)(xel such that 0 ^ d'n ^ 1 for all a e L Let <p(u) = inf(w, 1) for 0 ^ u < + oo. Then <p(0) = 0, and (p is an increas- ing function on [0, 4- oo[ and satisfies the inequality (p(u + v) :g <p(w) + <p(tf). (This last assertion is clear if u> 1 or i; > 1 or if w + i? ^ 1; and if u ^ 1 and v <£ 1 and i/ + t; > 1 we have <p(w + u) = 1 < <p(«) + <p(f).) It follows now from (12.4.1) that d'a = <p ° da is a pseudo-distance on E, for each a e I. That the topologies defined by the families (da) and (d£) are the same follows from (12.2.1) and the fact that, when 0<r< 1, the relations dx(x,y)<r and ^«(X JO < r are equivalent. (12.4.6) A Hausdorff uniformizable space E, w/zo^ topology can be defined by a sequence (dn) of pseudo-distances, is metrizable. Without loss of generality we may assume that the sequence (dn) is infinite, and by (12.4.5) that 0 ^ dn ^ 1 for all n. Then the series (1 2.4.6.1 ) d(x, y) « 1 rf^jc, y) + ~ d2(x, y) + ~- + ± dn(x, y) + -- converges for all pairs (x, y) of elements of E, and d is a pseudo-distance on E (12.4.1). Moreover, d is a distance, because d(x, j>) = 0 implies that dn(x9 7) = 0 f°r a^ ^> anc^ therefore that x = y because E is Hausdorff (12.4.4). If B0(:x;; r) is the open ball with center x and radius r with respect to the dis- tance d, then it follows immediately from (1 2.4.6.1 ) that BO(JC ; r) c E(x ; », 2V) for all n. Conversely, if n is so large that 2n~l ^ 1/r, then we have for all x, 7 in E, and therefore B(*; (1,2,..., n), (ir, . . . , ir)) cz BO(JC, r). By (12.2.1), the proof is complete. (12.4.7) Let E be a topological space and (Un) a finite or denumerable open covering ofE such that the subspaces Uw are separable and metrizable. Then E is separable and metrizable.? ^/2) of x and 7, respectively, are disjoint, by virtue of the triangle